Sparring

In his published paper, Shen Qi used PLAN-A to prove the Wash Conjecture.

Suppose (X, Y) is a solution to the equation (t+1)X^4-tY^2=1, satisfying Y>1, (x, y) is the corresponding adjoint solution, and N=√x^2+y^2t. Then for some positive integer t0 that satisfies t0|t and t0^2≤t, we have P(x, y)=t0^2.

This is the core step in proving the Wash Conjecture. We define r0 as a positive integer that satisfies (e^2.37ε2/8)^1-r0≤∣fq∣≤(e^2.37ε2/8)^-r0. Shen Qi used PLAN-A in his paper.

In PLAN-A, Shen Qi r0=1, ±B1q≠A1p and 2∣fq∣(e^2.37ε2/8)<1.

He obtained △=K(±B1q-pA1)≠0, thus finally proving that the equation (t+1)X^4-tY^2=1 does not have two sets of positive integer solutions (Xi, Yi) (i=1,2), Y2>Y1>1 satisfying |±√-1(xi-yi√-t)/(xi+yi√-t)-X^1/4|<1/8.

Therefore, Mr. Wash's conjecture made 37 years ago was correct.

This conjecture was confirmed by a 21-year-old Chinese student studying abroad.

Shen Qi thus received some honors and awards, and began to make a name for himself in Mathematics Community Chinese and American Mathematics Community.

The set of mathematical symbols that Old Wu just wrote down represents PLAN-B, which is another way of proving the core steps of the Wash Conjecture.

It turned out that Old Wu had read my paper published in the Journal of the American Mathematical Society. Shen Qi understood.

In fact, Shen Qi only recently grasped the concept of PLAN-B, thanks to the persistent questioning from the group of mathematical giants at Princeton.

However, Shen Qi had already published his paper based on PLAN-A by then.

For him, Plan-B was a supplement rather than a necessity, so Shen Qi did not immediately refine the specific operational plan for Plan-B, leaving it as a thought in mind.

Then, Shen Qi was informed that he had won the Chern Prize in Mathematics. During this special period, he was even less able to change the already published Plan A.

A few days ago, Shen Qi upgraded his math level to level 10, and he fully comprehended PLAN-B in a virtual scenario in his mind.

So, Old Wu wanted to discuss Plan-B with me, but he didn't want to explain it too clearly; everything was left unsaid... Shen Qi walked to the whiteboard, picked up a marker, and wrote:

N2≥N1^7/6t^2

After finishing writing, Shen Qi humbly asked for guidance, "Please give me your advice, Old Wu."

"You're young, but pragmatic. I like pragmatic young people." Old Wu smiled, casually wiping away Shen Qi's ≥, and then cubed N2.

Therefore, Shen Qi's answer N2≥N1^7/6t^2 changed to "N2^3 blank N1^7/6t^2".

"Old Wu is indeed a cut above the rest." Shen Qi cupped his hands in admiration, then added, "But I still have a way out."

Shen Qi filled in the blank with ≤, added N1 before N2^3, then erased N1^7/6t^2, replacing it with 54B^2t^1.5

Therefore, the latest answer becomes:

N1 N2^3≤54B^2t^1.5

"Young people are quick-witted, have broad perspectives, and are truly formidable," Old Wu said with a smile, then wrote down a very complex formula:

2t2^2/√t+1N1^4(N2/N1)^4=...8/(e^0.99ε1)^2(3N2/N1)

"Hahaha!" Shen Qi laughed heartily, giving a thumbs up. "I'm impressed, I'm impressed! Old Wu is indeed a towering figure, his words can reduce enemy ships to ashes with a mere laugh."

"Is there a solution?" Old Wu asked, awaiting Shen Qi's answer.

"There is still one more strategy: burn the boats and take the plunge." Shen Qi couldn't help but admire the academician, who was indeed a highly skilled individual.

Then Shen Qi picked up his pen and wrote down a more complex formula:

∣(4B√-t+4A)(u+v√-t)^4-(4B√-t-4A)(uv√-t)^4∣...=8N1^8t2^2, t2<√t

The other people in the meeting room were either deep in thought or looked completely bewildered.

"Hahaha!" Academician Wu laughed heartily and said, "Different paths, same destination."

"Hahaha!" Shen Qi laughed heartily. Only Academician Wu understood him. "Different paths, same destination."

The two mathematicians, one old and one young, admired each other and seemed to have become close friends despite their age difference.

The people in the room looked at each other, not daring to speak, unsure of what to say, only feeling that this was going to be a high-level discussion, extremely valuable for research.

"Just wipe it off, it's really not very useful," Old Wu suddenly shook his head and said to Shen Qi.

"It's really useless. It's enough to write one word for fennel for fennel bean." Shen Qi erased all the writing on the whiteboard, and his ideological level further improved.

"This..." The others were speechless. What were you two doing? Writing, erasing, writing, erasing completely, trying to solve a riddle?

"Professor Sun, what exactly happened between Shen Qi and Academician Wu?" Zhou Yu'an asked in a low voice, his curiosity piqued.

"Heavenly secrets cannot be revealed," Sun Erxiong said mysteriously.

"Well then, that concludes my presentation for today. Thank you to all the experts for participating. Now, let's invite Academician Wu to say a few words." Shen Qi felt it was time to wrap things up, and as is customary, the leader was to give a concluding speech.

"Three sentences: When studying mathematics, don't engage in empty talk; maintain a calm and patient mindset; learning is endless; be careful in your words and actions. Meeting adjourned." After saying this, Old Wu left with his hands behind his back. At the door, he turned back and said, "Shen Qi, come out with me for a moment."

Shen Qi nodded, put his hands behind his back, and left.

The room was left with a room full of people discussing it.

"Shen Qi is about to receive individual guidance from Academician Wu." Zhou Yu'an didn't quite understand what Shen Qi was showing off about, but he could understand that Shen Qi must have shown off and shocked everyone.

"Shen Qi is young and promising."

"This young man is quite good; he's talented, unassuming, and knows how to control his emotions."

"Such a composed and measured young man is rare these days."

"Those who can pursue advanced studies in the Princeton Mathematics Department are truly talented."

Everyone praised Shen Qi highly, and a new star was rising in the Chinese Mathematics Community.

In a corner of the University of Hong Kong, Academician Wu and Shen Qi had a private conversation.

"I've read your paper published in the Journal of the American Mathematical Society. Let a and b be positive integers. Then the Diophantine equation written by Wash has at most two sets of positive integer solutions. Shen Qi, your proof method is the most perfect. I just had a sudden inspiration. Let me indulge in a bit of youthful whimsy," Academician Wu said.

"Thank you for your guidance, Old Wu."

"It's not really guidance, I just wanted to verify an idea with you."

In short, Academician Wu found some inspiration from Shen Qi's PLAN-A, so he came up with PLAN-B and played a game with Shen Qi.

Academician Wu's PLAN-B, which evoked a youthful passion in his old age, is exactly the same PLAN-B that Shen Qi recently came to understand.

Their ideas collided, and they reached a consensus: Plan B wasn't so great after all; Plan A was much better.

The old man and the young man playing games in the world, which outsiders see as a high-end show, is just a little game in the eyes of Shen Qi and Academician Wu.

As one progresses through mathematics to a certain level, true friends become increasingly rare, and feelings of loneliness intensify. Shen Qi remarked, "Actually, I've long admired Old Wu, and today I was fortunate enough to exchange ideas with Old Wu in person, from which I benefited immensely."

"What are your plans for the future? Will you come back?" Academician Wu asked.

Shen Qi nodded. "Of course, after I get my PhD in mathematics from Princeton, I will return to China. My roots are in China."

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